Image Captioning with LSTMs¶
In the previous exercise you implemented a vanilla RNN and applied it to image captioning. In this notebook you will implement the LSTM update rule and use it for image captioning.
# As usual, a bit of setup
import time, os, json
import numpy as np
import matplotlib.pyplot as plt
from cs231n.gradient_check import eval_numerical_gradient, eval_numerical_gradient_array
from cs231n.rnn_layers import *
from cs231n.captioning_solver import CaptioningSolver
from cs231n.classifiers.rnn import CaptioningRNN
from cs231n.coco_utils import load_coco_data, sample_coco_minibatch, decode_captions
from cs231n.image_utils import image_from_url
%matplotlib inline
plt.rcParams['figure.figsize'] = (10.0, 8.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'
# for auto-reloading external modules
# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2
def rel_error(x, y):
""" returns relative error """
return np.max(np.abs(x - y) / (np.maximum(1e-8, np.abs(x) + np.abs(y))))
Load MS-COCO data¶
As in the previous notebook, we will use the Microsoft COCO dataset for captioning.
# Load COCO data from disk; this returns a dictionary
# We'll work with dimensionality-reduced features for this notebook, but feel
# free to experiment with the original features by changing the flag below.
data = load_coco_data(pca_features=True)
# Print out all the keys and values from the data dictionary
for k, v in data.items():
if type(v) == np.ndarray:
print( k, type(v), v.shape, v.dtype)
else:
print( k, type(v), len(v))
LSTM¶
If you read recent papers, you'll see that many people use a variant on the vanialla RNN called Long-Short Term Memory (LSTM) RNNs. Vanilla RNNs can be tough to train on long sequences due to vanishing and exploding gradiants caused by repeated matrix multiplication. LSTMs solve this problem by replacing the simple update rule of the vanilla RNN with a gating mechanism as follows.
Similar to the vanilla RNN, at each timestep we receive an input $x_t\in\mathbb{R}^D$ and the previous hidden state $h_{t-1}\in\mathbb{R}^H$; the LSTM also maintains an $H$-dimensional cell state, so we also receive the previous cell state $c_{t-1}\in\mathbb{R}^H$. The learnable parameters of the LSTM are an input-to-hidden matrix $W_x\in\mathbb{R}^{4H\times D}$, a hidden-to-hidden matrix $W_h\in\mathbb{R}^{4H\times H}$ and a bias vector $b\in\mathbb{R}^{4H}$.
At each timestep we first compute an activation vector $a\in\mathbb{R}^{4H}$ as $a=W_xx_t + W_hh_{t-1}+b$. We then divide this into four vectors $a_i,a_f,a_o,a_g\in\mathbb{R}^H$ where $a_i$ consists of the first $H$ elements of $a$, $a_f$ is the next $H$ elements of $a$, etc. We then compute the input gate $g\in\mathbb{R}^H$, forget gate $f\in\mathbb{R}^H$, output gate $o\in\mathbb{R}^H$ and block input $g\in\mathbb{R}^H$ as
$$ \begin{align*} i = \sigma(a_i) \hspace{2pc} f = \sigma(a_f) \hspace{2pc} o = \sigma(a_o) \hspace{2pc} g = \tanh(a_g) \end{align*} $$where $\sigma$ is the sigmoid function and $\tanh$ is the hyperbolic tangent, both applied elementwise.
Finally we compute the next cell state $c_t$ and next hidden state $h_t$ as
$$ c_{t} = f\odot c_{t-1} + i\odot g \hspace{4pc} h_t = o\odot\tanh(c_t) $$where $\odot$ is the elementwise product of vectors.
In the rest of the notebook we will implement the LSTM update rule and apply it to the image captioning task.
LSTM: step forward¶
Implement the forward pass for a single timestep of an LSTM in the lstm_step_forward function in the file cs231n/rnn_layers.py. This should be similar to the rnn_step_forward function that you implemented above, but using the LSTM update rule instead.
Once you are done, run the following to perform a simple test of your implementation. You should see errors around 1e-8 or less.
N, D, H = 3, 4, 5
x = np.linspace(-0.4, 1.2, num=N*D).reshape(N, D)
prev_h = np.linspace(-0.3, 0.7, num=N*H).reshape(N, H)
prev_c = np.linspace(-0.4, 0.9, num=N*H).reshape(N, H)
Wx = np.linspace(-2.1, 1.3, num=4*D*H).reshape(D, 4 * H)
Wh = np.linspace(-0.7, 2.2, num=4*H*H).reshape(H, 4 * H)
b = np.linspace(0.3, 0.7, num=4*H)
next_h, next_c, cache = lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)
expected_next_h = np.asarray([
[ 0.24635157, 0.28610883, 0.32240467, 0.35525807, 0.38474904],
[ 0.49223563, 0.55611431, 0.61507696, 0.66844003, 0.7159181 ],
[ 0.56735664, 0.66310127, 0.74419266, 0.80889665, 0.858299 ]])
expected_next_c = np.asarray([
[ 0.32986176, 0.39145139, 0.451556, 0.51014116, 0.56717407],
[ 0.66382255, 0.76674007, 0.87195994, 0.97902709, 1.08751345],
[ 0.74192008, 0.90592151, 1.07717006, 1.25120233, 1.42395676]])
print( 'next_h error: ', rel_error(expected_next_h, next_h))
print( 'next_c error: ', rel_error(expected_next_c, next_c))
LSTM: step backward¶
Implement the backward pass for a single LSTM timestep in the function lstm_step_backward in the file cs231n/rnn_layers.py. Once you are done, run the following to perform numeric gradient checking on your implementation. You should see errors around 1e-8 or less.
N, D, H = 4, 5, 6
x = np.random.randn(N, D)
prev_h = np.random.randn(N, H)
prev_c = np.random.randn(N, H)
Wx = np.random.randn(D, 4 * H)
Wh = np.random.randn(H, 4 * H)
b = np.random.randn(4 * H)
next_h, next_c, cache = lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)
dnext_h = np.random.randn(*next_h.shape)
dnext_c = np.random.randn(*next_c.shape)
fx_h = lambda x: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[0]
fh_h = lambda h: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[0]
fc_h = lambda c: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[0]
fWx_h = lambda Wx: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[0]
fWh_h = lambda Wh: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[0]
fb_h = lambda b: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[0]
fx_c = lambda x: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[1]
fh_c = lambda h: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[1]
fc_c = lambda c: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[1]
fWx_c = lambda Wx: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[1]
fWh_c = lambda Wh: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[1]
fb_c = lambda b: lstm_step_forward(x, prev_h, prev_c, Wx, Wh, b)[1]
num_grad = eval_numerical_gradient_array
dx_num = num_grad(fx_h, x, dnext_h) + num_grad(fx_c, x, dnext_c)
dh_num = num_grad(fh_h, prev_h, dnext_h) + num_grad(fh_c, prev_h, dnext_c)
dc_num = num_grad(fc_h, prev_c, dnext_h) + num_grad(fc_c, prev_c, dnext_c)
dWx_num = num_grad(fWx_h, Wx, dnext_h) + num_grad(fWx_c, Wx, dnext_c)
dWh_num = num_grad(fWh_h, Wh, dnext_h) + num_grad(fWh_c, Wh, dnext_c)
db_num = num_grad(fb_h, b, dnext_h) + num_grad(fb_c, b, dnext_c)
dx, dh, dc, dWx, dWh, db = lstm_step_backward(dnext_h, dnext_c, cache)
print( 'dx error: ', rel_error(dx_num, dx))
print( 'dh error: ', rel_error(dh_num, dh))
print( 'dc error: ', rel_error(dc_num, dc))
print( 'dWx error: ', rel_error(dWx_num, dWx))
print( 'dWh error: ', rel_error(dWh_num, dWh))
print( 'db error: ', rel_error(db_num, db))
LSTM: forward¶
In the function lstm_forward in the file cs231n/rnn_layers.py, implement the lstm_forward function to run an LSTM forward on an entire timeseries of data.
When you are done run the following to check your implementation. You should see an error around 1e-7.
N, D, H, T = 2, 5, 4, 3
x = np.linspace(-0.4, 0.6, num=N*T*D).reshape(N, T, D)
h0 = np.linspace(-0.4, 0.8, num=N*H).reshape(N, H)
Wx = np.linspace(-0.2, 0.9, num=4*D*H).reshape(D, 4 * H)
Wh = np.linspace(-0.3, 0.6, num=4*H*H).reshape(H, 4 * H)
b = np.linspace(0.2, 0.7, num=4*H)
h, cache = lstm_forward(x, h0, Wx, Wh, b)
expected_h = np.asarray([
[[ 0.01764008, 0.01823233, 0.01882671, 0.0194232 ],
[ 0.11287491, 0.12146228, 0.13018446, 0.13902939],
[ 0.31358768, 0.33338627, 0.35304453, 0.37250975]],
[[ 0.45767879, 0.4761092, 0.4936887, 0.51041945],
[ 0.6704845, 0.69350089, 0.71486014, 0.7346449 ],
[ 0.81733511, 0.83677871, 0.85403753, 0.86935314]]])
print( 'h error: ', rel_error(expected_h, h))
LSTM: backward¶
Implement the backward pass for an LSTM over an entire timeseries of data in the function lstm_backward in the file cs231n/rnn_layers.py. When you are done run the following to perform numeric gradient checking on your implementation. You should see errors around 1e-8 or less.
from cs231n.rnn_layers import lstm_forward, lstm_backward
N, D, T, H = 2, 3, 10, 6
x = np.random.randn(N, T, D)
h0 = np.random.randn(N, H)
Wx = np.random.randn(D, 4 * H)
Wh = np.random.randn(H, 4 * H)
b = np.random.randn(4 * H)
out, cache = lstm_forward(x, h0, Wx, Wh, b)
dout = np.random.randn(*out.shape)
dx, dh0, dWx, dWh, db = lstm_backward(dout, cache)
fx = lambda x: lstm_forward(x, h0, Wx, Wh, b)[0]
fh0 = lambda h0: lstm_forward(x, h0, Wx, Wh, b)[0]
fWx = lambda Wx: lstm_forward(x, h0, Wx, Wh, b)[0]
fWh = lambda Wh: lstm_forward(x, h0, Wx, Wh, b)[0]
fb = lambda b: lstm_forward(x, h0, Wx, Wh, b)[0]
dx_num = eval_numerical_gradient_array(fx, x, dout)
dh0_num = eval_numerical_gradient_array(fh0, h0, dout)
dWx_num = eval_numerical_gradient_array(fWx, Wx, dout)
dWh_num = eval_numerical_gradient_array(fWh, Wh, dout)
db_num = eval_numerical_gradient_array(fb, b, dout)
print( 'dx error: ', rel_error(dx_num, dx))
print( 'dh0 error: ', rel_error(dh0_num, dh0))
print( 'dWx error: ', rel_error(dWx_num, dWx))
print( 'dWh error: ', rel_error(dWh_num, dWh))
print( 'db error: ', rel_error(db_num, db))
LSTM captioning model¶
Now that you have implemented an LSTM, update the implementation of the loss method of the CaptioningRNN class in the file cs231n/classifiers/rnn.py to handle the case where self.cell_type is lstm. This should require adding less than 10 lines of code.
Once you have done so, run the following to check your implementation. You should see a difference of less than 1e-10.
N, D, W, H = 10, 20, 30, 40
word_to_idx = {'<NULL>': 0, 'cat': 2, 'dog': 3}
V = len(word_to_idx)
T = 13
model = CaptioningRNN(word_to_idx,
input_dim=D,
wordvec_dim=W,
hidden_dim=H,
cell_type='lstm',
dtype=np.float64)
# Set all model parameters to fixed values
for k, v in model.params.items():
model.params[k] = np.linspace(-1.4, 1.3, num=v.size).reshape(*v.shape)
features = np.linspace(-0.5, 1.7, num=N*D).reshape(N, D)
captions = (np.arange(N * T) % V).reshape(N, T)
loss, grads = model.loss(features, captions)
expected_loss = 9.82445935443
print( 'loss: ', loss)
print( 'expected loss: ', expected_loss)
print( 'difference: ', abs(loss - expected_loss))
Overfit LSTM captioning model¶
Run the following to overfit an LSTM captioning model on the same small dataset as we used for the RNN above.
small_data = load_coco_data(max_train=50)
small_lstm_model = CaptioningRNN(
cell_type='lstm',
word_to_idx=data['word_to_idx'],
input_dim=data['train_features'].shape[1],
hidden_dim=512,
wordvec_dim=256,
dtype=np.float32,
)
small_lstm_solver = CaptioningSolver(small_lstm_model, small_data,
update_rule='adam',
num_epochs=50,
batch_size=25,
optim_config={
'learning_rate': 5e-3,
},
lr_decay=0.995,
verbose=True, print_every=10,
)
small_lstm_solver.train()
# Plot the training losses
plt.plot(small_lstm_solver.loss_history)
plt.xlabel('Iteration')
plt.ylabel('Loss')
plt.title('Training loss history')
plt.show()
LSTM test-time sampling¶
Modify the sample method of the CaptioningRNN class to handle the case where self.cell_type is lstm. This should take fewer than 10 lines of code.
When you are done run the following to sample from your overfit LSTM model on some training and validation set samples.
for split in ['train', 'val']:
minibatch = sample_coco_minibatch(small_data, split=split, batch_size=2)
gt_captions, features, urls = minibatch
gt_captions = decode_captions(gt_captions, data['idx_to_word'])
sample_captions = small_lstm_model.sample(features)
sample_captions = decode_captions(sample_captions, data['idx_to_word'])
for gt_caption, sample_caption, url in zip(gt_captions, sample_captions, urls):
plt.imshow(image_from_url(url))
plt.title('%s\n%s\nGT:%s' % (split, sample_caption, gt_caption))
plt.axis('off')
plt.show()
Train a good captioning model!¶
Using the pieces you have implemented in this and the previous notebook, try to train a captioning model that gives decent qualitative results (better than the random garbage you saw with the overfit models) when sampling on the validation set. You can subsample the training set if you want; we just want to see samples on the validatation set that are better than random.
Don't spend too much time on this part; we don't have any explicit accuracy thresholds you need to meet.
small_data = load_coco_data(max_train=500)
small_lstm_model = CaptioningRNN(
cell_type='lstm',
word_to_idx=data['word_to_idx'],
input_dim=data['train_features'].shape[1],
hidden_dim=512,
wordvec_dim=256,
dtype=np.float32,
)
small_lstm_solver = CaptioningSolver(small_lstm_model, small_data,
update_rule='adam',
num_epochs=25,
batch_size=25,
optim_config={
'learning_rate': 5e-3,
},
lr_decay=0.995,
verbose=True, print_every=10,
)
# training cut added to train with validation split
small_lstm_solver.train(best=True, early_stop=True)
# Plot the training losses
plt.plot(small_lstm_solver.loss_history)
plt.xlabel('Iteration')
plt.ylabel('Loss')
plt.title('Training loss history')
plt.show()
for split in ['train', 'val']:
minibatch = sample_coco_minibatch(small_data, split=split, batch_size=2)
gt_captions, features, urls = minibatch
gt_captions = decode_captions(gt_captions, data['idx_to_word'])
sample_captions = small_lstm_model.sample(features)
sample_captions = decode_captions(sample_captions, data['idx_to_word'])
for gt_caption, sample_caption, url in zip(gt_captions, sample_captions, urls):
plt.imshow(image_from_url(url))
plt.title('%s\n%s\nGT:%s' % (split, sample_caption, gt_caption))
plt.axis('off')
plt.show()
